Chebyshev function

The second Chebyshev function can be seen to be related to the first by writing it as

${\displaystyle \psi (x)=\sum _{p\leq x}k\log p}$

where k is the unique integer such that p^k ≤ x and x < p^k + 1. The values of k are given in OEIS: A206722. A more direct relationship is given by

${\displaystyle \psi (x)=\sum _{n=1}^{\infty }\vartheta {\big (}x^{\frac {1}{n}}{\big )}.}$

Note that this last sum has only a finite number of non-vanishing terms, as

${\displaystyle \vartheta {\big (}x^{\frac {1}{n}}{\big )}=0\quad {\text{for}}\quad n>\log _{2}x={\frac {\log x}{\log 2}}.}$

The second Chebyshev function is the logarithm of the least common multiple of the integers from 1 to n.

${\displaystyle \operatorname {lcm} (1,2,\dots ,n)=e^{\psi (n)}.}$

Values of lcm(1, 2, ..., n) for the integer variable n are given at OEIS: A003418.

Source:[3]

The following theorem relates the two quotients ${\displaystyle {\frac {\psi (x)}{x}}}$ and ${\displaystyle {\frac {\vartheta (x)}{x}}}$.

Theorem: For ${\displaystyle x>0}$, we have

${\displaystyle 0\leq {\frac {\psi (x)}{x}}-{\frac {\vartheta (x)}{x}}\leq {\frac {(\log x)^{2}}{2{\sqrt {x}}\log 2}}.}$

Note: This inequality implies that

${\displaystyle \lim _{x\to \infty }\!\left({\frac {\psi (x)}{x}}-{\frac {\vartheta (x)}{x}}\right)\!=0.}$

In other words, if one of the ${\displaystyle \psi (x)/x}$ or ${\displaystyle \vartheta (x)/x}$ tends to a limit then so does the other, and the two limits are equal.

Proof: Since ${\displaystyle \psi (x)=\sum _{n\leq \log _{2}x}\vartheta (x^{1/n})}$, we find that

${\displaystyle 0\leq \psi (x)-\vartheta (x)=\sum _{2\leq n\leq \log _{2}x}\vartheta (x^{1/n}).}$

But from the definition of ${\displaystyle \vartheta (x)}$ we have the trivial inequality

${\displaystyle \vartheta (x)\leq \sum _{p\leq x}\log x\leq x\log x}$

so

${\displaystyle {\begin{aligned}0\leq \psi (x)-\vartheta (x)&\leq \sum _{2\leq n\leq \log _{2}x}x^{1/n}\log(x^{1/n})\\&\leq (\log _{2}x){\sqrt {x}}\log {\sqrt {x}}\\&={\frac {\log x}{\log 2}}{\frac {\sqrt {x}}{2}}\log x\\&={\frac {{\sqrt {x}}\,(\log x)^{2}}{2\log 2}}.\end{aligned}}}$

Lastly, divide by ${\displaystyle x}$ to obtain the inequality in the theorem.

The following bounds are known for the Chebyshev functions: (in these formulas p_k is the kth prime number; p₁ = 2, p₂ = 3, etc.)

${\displaystyle {\begin{aligned}\vartheta (p_{k})&\geq k\left(\log k+\log \log k-1+{\frac {\log \log k-2.050735}{\log k}}\right)&&{\text{for }}k\geq 10^{11},\\[8px]\vartheta (p_{k})&\leq k\left(\log k+\log \log k-1+{\frac {\log \log k-2}{\log k}}\right)&&{\text{for }}k\geq 198,\\[8px]|\vartheta (x)-x|&\leq 0.006788\,{\frac {x}{\log x}}&&{\text{for }}x\geq 10\,544\,111,\\[8px]|\psi (x)-x|&\leq 0.006409\,{\frac {x}{\log x}}&&{\text{for }}x\geq e^{22},\\[8px]0.9999{\sqrt {x}}&<\psi (x)-\vartheta (x)<1.00007{\sqrt {x}}+1.78{\sqrt[{3}]{x}}&&{\text{for }}x\geq 121.\end{aligned}}}$

Furthermore, under the Riemann hypothesis,

${\displaystyle {\begin{aligned}|\vartheta (x)-x|&=O{\Big (}x^{{\frac {1}{2}}+\varepsilon }{\Big )}\\|\psi (x)-x|&=O{\Big (}x^{{\frac {1}{2}}+\varepsilon }{\Big )}\end{aligned}}}$

for any ε > 0.

Upper bounds exist for both ϑ  (x) and ψ (x) such that[4]

${\displaystyle {\begin{aligned}\vartheta (x)&<1.000028x\\\psi (x)&<1.03883x\end{aligned}}}$

for any x > 0.

An explanation of the constant 1.03883 is given at OEIS: A206431.

In 1895, Hans Carl Friedrich von Mangoldt proved an explicit expression for ψ (x) as a sum over the nontrivial zeros of the Riemann zeta function:

${\displaystyle \psi _{0}(x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-{\frac {\zeta '(0)}{\zeta (0)}}-{\tfrac {1}{2}}\log(1-x^{-2}).}$

(The numerical value of ζ′ (0)/ζ (0) is log(2π).) Here ρ runs over the nontrivial zeros of the zeta function, and ψ₀ is the same as ψ, except that at its jump discontinuities (the prime powers) it takes the value halfway between the values to the left and the right:

${\displaystyle \psi _{0}(x)={\frac {1}{2}}\!\left(\sum _{n\leq x}\Lambda (n)+\sum _{n<x}\Lambda (n)\right)={\begin{cases}\psi (x)-{\tfrac {1}{2}}\Lambda (x)&x=2,3,4,5,7,8,9,11,13,16,\dots \\[5px]\psi (x)&{\mbox{otherwise.}}\end{cases}}}$

From the Taylor series for the logarithm, the last term in the explicit formula can be understood as a summation of x^ω/ω over the trivial zeros of the zeta function, ω = −2, −4, −6, ..., i.e.

${\displaystyle \sum _{k=1}^{\infty }{\frac {x^{-2k}}{-2k}}={\tfrac {1}{2}}\log \left(1-x^{-2}\right).}$

Similarly, the first term, x = x¹/1, corresponds to the simple pole of the zeta function at 1. It being a pole rather than a zero accounts for the opposite sign of the term.

A theorem due to Erhard Schmidt states that, for some explicit positive constant K, there are infinitely many natural numbers x such that

${\displaystyle \psi (x)-x<-K{\sqrt {x}}}$

and infinitely many natural numbers x such that

${\displaystyle \psi (x)-x>K{\sqrt {x}}.}$

In little-o notation, one may write the above as

${\displaystyle \psi (x)-x\neq o\left({\sqrt {x}}\,\right).}$

Hardy and Littlewood prove the stronger result, that

${\displaystyle \psi (x)-x\neq o\left({\sqrt {x}}\,\log \log \log x\right).}$

The first Chebyshev function is the logarithm of the primorial of x, denoted x #:

${\displaystyle \vartheta (x)=\sum _{p\leq x}\log p=\log \prod _{p\leq x}p=\log \left(x\#\right).}$

This proves that the primorial x # is asymptotically equal to e^{(1  + o(1))x}, where "o" is the little-o notation (see big O notation) and together with the prime number theorem establishes the asymptotic behavior of p_n #.

The Chebyshev function can be related to the prime-counting function as follows. Define

${\displaystyle \Pi (x)=\sum _{n\leq x}{\frac {\Lambda (n)}{\log n}}.}$

Then

${\displaystyle \Pi (x)=\sum _{n\leq x}\Lambda (n)\int _{n}^{x}{\frac {dt}{t\log ^{2}t}}+{\frac {1}{\log x}}\sum _{n\leq x}\Lambda (n)=\int _{2}^{x}{\frac {\psi (t)\,dt}{t\log ^{2}t}}+{\frac {\psi (x)}{\log x}}.}$

The transition from Π to the prime-counting function, π, is made through the equation

${\displaystyle \Pi (x)=\pi (x)+{\tfrac {1}{2}}\pi \left({\sqrt {x}}\,\right)+{\tfrac {1}{3}}\pi \left({\sqrt[{3}]{x}}\,\right)+\cdots }$

Certainly π (x) ≤ x, so for the sake of approximation, this last relation can be recast in the form

${\displaystyle \pi (x)=\Pi (x)+O\left({\sqrt {x}}\,\right).}$

The Riemann hypothesis states that all nontrivial zeros of the zeta function have real part 1/2. In this case, |x^ρ| = √x, and it can be shown that

${\displaystyle \sum _{\rho }{\frac {x^{\rho }}{\rho }}=O\!\left({\sqrt {x}}\,\log ^{2}x\right).}$

By the above, this implies

${\displaystyle \pi (x)=\operatorname {li} (x)+O\!\left({\sqrt {x}}\,\log x\right).}$

Good evidence that the hypothesis could be true comes from the fact proposed by Alain Connes and others, that if we differentiate the von Mangoldt formula with respect to x we get x = e^u. Manipulating, we have the "Trace formula" for the exponential of the Hamiltonian operator satisfying

${\displaystyle \left.\zeta {\big (}{\tfrac {1}{2}}+i{\hat {H}}{\big )}\right|n\geq \zeta \!\left({\tfrac {1}{2}}+iE_{n}\right)=0,}$

and

${\displaystyle \sum _{n}e^{iuE_{n}}=Z(u)=e^{\frac {u}{2}}-e^{-{\frac {u}{2}}}{\frac {d\psi _{0}}{du}}-{\frac {e^{\frac {u}{2}}}{e^{3u}-e^{u}}}=\operatorname {Tr} \!{\big (}e^{iu{\hat {H}}}{\big )},}$

where the "trigonometric sum" can be considered to be the trace of the operator (statistical mechanics) e^iuĤ, which is only true if ρ = 1/2 + iE(n).

Using the semiclassical approach the potential of H = T + V satisfies:

${\displaystyle {\frac {Z(u)u^{\frac {1}{2}}}{\sqrt {\pi }}}\sim \int _{-\infty }^{\infty }e^{i\left(uV(x)+{\frac {\pi }{4}}\right)}\,dx}$

with Z (u) → 0 as u → ∞.

solution to this nonlinear integral equation can be obtained (among others) by

${\displaystyle V^{-1}(x)\approx {\sqrt {4\pi }}\cdot {\frac {d^{\frac {1}{2}}}{dx^{\frac {1}{2}}}}N(x)}$

in order to obtain the inverse of the potential:

${\displaystyle \pi N(E)=\operatorname {Arg} \xi \left({\tfrac {1}{2}}+iE\right).}$

The difference of the smoothed Chebyshev function and x²/2 for x < 10⁶

The smoothing function is defined as

${\displaystyle \psi _{1}(x)=\int _{0}^{x}\psi (t)\,dt.}$

Obviously ${\displaystyle \psi _{1}(x)\sim {\frac {x^{2}}{2}}.}$

The Chebyshev function evaluated at x = e^t minimizes the functional

${\displaystyle J[f]=\int _{0}^{\infty }{\frac {f(s)\zeta '(s+c)}{\zeta (s+c)(s+c)}}\,ds-\int _{0}^{\infty }\!\!\!\int _{0}^{\infty }e^{-st}f(s)f(t)\,ds\,dt,}$

so

${\displaystyle f(t)=\psi (e^{t})e^{-ct}\quad {\text{for }}c>0.}$

• ^ Pierre Dusart, "Estimates of some functions over primes without R.H.". arXiv:1002.0442
• ^ Pierre Dusart, "Sharper bounds for ψ, θ, π, p_k", Rapport de recherche no. 1998-06, Université de Limoges. An abbreviated version appeared as "The kth prime is greater than k(log k + log log k − 1) for k ≥ 2", Mathematics of Computation, Vol. 68, No. 225 (1999), pp. 411–415.
• ^ Erhard Schmidt, "Über die Anzahl der Primzahlen unter gegebener Grenze", Mathematische Annalen, 57 (1903), pp. 195–204.
• ^ G .H. Hardy and J. E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Mathematica, 41 (1916) pp. 119–196.
• ^ Davenport, Harold (2000). In Multiplicative Number Theory. Springer. p. 104. ISBN 0-387-95097-4. Google Book Search.

• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001

• Weisstein, Eric W. "Chebyshev functions". MathWorld.
• "Mangoldt summatory function". PlanetMath.
• "Chebyshev functions". PlanetMath.
• Riemann's Explicit Formula, with images and movies